Color Energy of Generalised Complements of a Graph Swati Nayak, Sabitha D’Souza∗, Gowtham H

IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_01 ______________________________________________________________________________________ Color Energy of Generalised Complements of a Graph Swati Nayak, Sabitha D’Souza∗, Gowtham H. J. and Pradeep G. Bhat, Abstract—The color energy of a graph is defined as sum A coloring of graph G is a coloring of its vertices such of absolute color eigenvalues of graph, denoted by Ec(G). Let that no two adjacent vertices receive the same color. The Gc = (V; E) be a color graph and P = fV1;V2;:::;Vkg be a minimum number of colors needed for coloring a graph G partition of V of order k ≥ 1. The k -color complement fG gP c k is called chromatic number, denoted by χ(G). of Gc is defined as follows: For all Vi and Vj in P , i 6= j, remove the edges between Vi The color matrix is as follows. If c(vi) is the color of vi, and Vj and add the edges which are not in Gc such that end then 8 1; if v ∼ v with c(v ) 6= c(v ); vertices have different colors. For each set Vr in the partition < i j i j P , remove the edges of Gc inside Vr, and add the edges of Gc aij = −1; if vi vj with c(vi) = c(vj); (the complement of Gc ) joining the vertices of Vr. The graph : P 0; otherwise fGcgk(i) thus obtained is called the k(i)− color complement The matrix thus obtained is the L-matrix of the colored G P V of c with respect to the partition of . In this paper we G A (G) characterize generalized color complements of some graphs. We graph denoted by c . Color energy of colored graph also compute color energy of generalised complements of star, is the sum of the absolute colored eigenvalues. complete, complete bipartite, crown, cocktail party, double star i.e., n and friendship graphs. X Ec(G) = λi: Index Terms—k−color complement, k(i)−color complement, i=1 color energy, color spectrum. In 2013, the concept of color energy of a graph was I. INTRODUCTION introduced by Chandrashekar Adiga et.al [12]. Let G = (V; E) be a colored graph. Then the complement Let G = (V; E) be a graph of order n. The complement of colored graph G denoted by G has the same coloring of of a graph G, denoted as G has the same vertex set as that c G with the following properties: of G, but two vertices are adjacent in G if and only if they are not adjacent in G. If G is isomorphic to G then G is • vi and vj are adjacent in Gc, if vi and vj are non- said to be self-complementary graph. For all notations and adjacent in G with c(vi) 6= c(vj). terminologies we refer [1], [2]. • vi and vj are non-adjacent in Gc, if vi and vj are non- The concept of energy of a graph was introduced by I. adjacent in G with c(vi) = c(vj). Gutman in the year 1978 as the sum of absolute eigenvalues • vi and vj are non-adjacent in Gc, if vi and vj are of graph G. More on energy of graphs we refer to [3], [4], adjacent in G. [5], [6], [7], [8], [9], [13], [14], [15]. In 1998 E. Sampathku- The paper is organised as follows. In section II, we mar et.al defined generalised complements of a graph i.e, introduce the concept of generalised color complements of P k−complement Gk of a graph G and k(i)−complement a graph. The preliminary results are shown in section III. P The characterisation of generalised color complements is Gk(i) of a graph G respectively [10],[11]. Let G = (V; E) be a graph and P = fV1;V2;:::;Vkg be presented in section IV. The color energy of generalised P complements of few families of graph is obtained in section a partition of V of order k ≥ 1. The k− complement Gk of G [10] is defined as follows: For all Vi and Vj in P , i 6= j, V. remove the edges between Vi and Vj and add the edges which are not in G. The graph G is k− self complementary (k − II. GENERALISED COLOR COMPLEMENTS OF A GRAPH s:c) with respect to P if GP =∼ G. Further, G is k−co−self k In this section we introduce a concept of generalised color complementary (k − co − s:c:) if GP =∼ G. k complements of a graph. For each set Vr in the partition P , remove the edges of G inside Vr, and add the edges of G joining the ver- Definition 1. Let Gc = (V; E) be a color graph and P = P tices of Vr. The graph Gk(i) thus obtained is called the fV1;V2;:::;Vkg be a partition of V of order k ≥ 1. P k(i)−complement [11] of G with respect to the partition P The k -color complement fGcgk of Gc is defined as follows: of V . The graph G is k(i)−self complementary (k(i) − s:c) For all Vi and Vj in P , i 6= j, remove the edges between P ∼ if Gk(i) = G for some partition P of order k. Further, G is Vi and Vj and add the edges which are not in Gc such that P ∼ end vertices have different colors. k(i)−co−self complementary (k(i) − co − s:c:) if Gk = G. Manuscript received March 10, 2020; revised July 10, 2020. • The graph Gc is k− self color complementary (k−s:c:c) P ∼ Department of Mathematics, Manipal Institute of Technology, with respect to P if fGcgk = Gc. Manipal Academy of Higher Education, Manipal, 576104 India • Further, Gc is k−co−self color complementary (k − e-mail: ([email protected], [email protected], P ∼ [email protected], [email protected]). co − s:c:c) if fGcgk = Gc. ∗ Corresponding author: SABITHA D’SOUZA, Department of Math- ematics, Manipal Institute of Technology, Manipal Academy of Higher Example 2. Let fV1;V2g be two partition of vertex set of Education, Manipal, 576104 India Cycle graph C4. Volume 50, Issue 4: December 2020 (Revised online publication: 4 December 2020) IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_01 ______________________________________________________________________________________ Proposition 8. [5] Let M; N; P; Q be matrices, and let M C C C C 1 2 1 2 be invertible. MN Let S= PQ Then det S = det M: det[Q − PM −1N]. If M and P C C 2 1 commute, then det S = det [MQ − PN]. C2 C1 P ()C ()C 4 c { 4 c} 2 IV. CHARACTERISATION OF GENERALISED COLOR Fig. 1. Colored cycle C4 and its 2− complement. COMPLEMENTS OF GRAPH Definition 3. For each set V in the partition P , remove r The following Proposition highlights the isomorphism the edges of G inside V , and add the edges of G (the c r c between the generalised color complements of a graph. complement of Gc ) joining the vertices of Vr. The graph P fGcgk(i) thus obtained is called the k(i)− color complement Proposition 9. For any Gc, of Gc with respect to the partition P of V . P ∼ P 1) (Gc)k = (Gc)k . • G k(i)− (k(i)− The graph c is self color complementary P ∼ P P ∼ 2) (Gc)k(i) = (Gc)k(i). s:c:c) if fGcgk(i) = Gc for some partition P of order k. Proof: • Further, Gc is k(i)−co−self color complementary P ∼ 1) Consider two vertices u and v in G . Then u ∼ v in (k(i) − co − s:c:c) if fGcg = Gc. c k(i) P P (Gc)k $ u v in (Gc)k . Example 4. $ u and v are in the same set in P , and are non-adjacent in Gc, or they are in different sets in P , and u ∼ v in Gc if c(u) 6= c(v). $ u and v are in same set in P , and u ∼ v in Gc C 1 C C C if c(u) 6= c(v), or they are in different sets in P , and 2 1 2 u v in Gc. P $ u ∼ v in (Gc)k if c(u) 6= c(v). P 2) Let u ∼ v in (Gc)k(i). P $ u v in (Gc)k(i). $ u and v are in same set in P , and u ∼ v in Gc C 2 C C C if c(u) 6= c(v), or they are in different sets in P , and 3 2 3 G { p u v in Gc. c Gc} 2 (i) $ u and v are in same set in P , u v in Gc, or they are in different sets in P , and u ∼ v in Gc if c(u) 6= c(v) . P P $ u ∼ v in (Gc)k(i) if c(u) 6= c(v). Fig. 2. Graph Gc and fGcg2(i). Definition 5. In a generalised color complement of graph As a consequence of Proposition 9, we have G, the indegree of vertex v denoted by i (v) is defined as d Corollary 10. For any graph G , number of edges incident to the vertex v inside the partite. c Outdegree of vertex v denoted by od(v) is defined as number P ∼ P ∼ 1) (Gc)k = Gc if and only if (Gc)k = Gc: of edges incident to the vertex v outside the partite.

Load more